Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches
@article{Gottwald2006UniversesOF,
title={Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches},
author={Siegfried Gottwald},
journal={Studia Logica},
year={2006},
volume={82},
pages={211-244}
}For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets.We discuss here the corresponding situation for fuzzy set theory.Our emphasis will be on various approaches toward (more or less naively formed)universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.What we give is a (critical)survey of quite a lot of…
39 Citations
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References
SHOWING 1-10 OF 97 REFERENCES
A development of set theory in fuzzy logic
- Computer Science
- 2003
This paper presents an axiomatic set theory FST ('Fuzzy Set Theory'), as a first-order theory within the framework of fuzzy logic in the style of [4], and shows that FST interprets ZF.
Terminological difficulties in fuzzy set theory - The case of "Intuitionistic Fuzzy Sets"
- Computer ScienceFuzzy Sets Syst.
- 2005
Fuzzy logic and fuzzy set theory
- Philosophy, Computer ScienceArch. Math. Log.
- 1992
An intuitionistic logic and intuitionistic set theory based on Gentzen's intuitionistic predicate logic with additional axioms and inference rules are presented, which asserts that the truth value set is linearly ordered and dense cHa.
A set theory within fuzzy logic
- Computer ScienceProceedings 31st IEEE International Symposium on Multiple-Valued Logic
- 2001
This paper proposes a possibility of developing an axiomatic set theory, as first-order theory within the framework of fuzzy logic in the style of Hajek's Basic fuzzy logic BL, and shows the nontriviality of the theory.
Set-valued set theory. II
- MathematicsNotre Dame J. Formal Log.
- 1975
This section needs to define the other concepts common in elementary classical set theory and then to verify that the sets given by the various definitions have the usual properties, including existence.